Complete Tripartite Graphs and their Competition Numbers Jaromy - - PowerPoint PPT Presentation

Complete Tripartite Graphs and their Competition Numbers

Jaromy Kuhl

Department of Mathematics and Statistics University of West Florida

Jaromy Kuhl (UWF) Competition Numbers 1 / 16

Competition Graphs

Definition 1 Let D = (V, A) be a digraph. The competition graph of D is the simple graph G = (V, E) where {u, v} ∈ E if and only if N+(u) ∩ N+(v) = ∅.

Jaromy Kuhl (UWF) Competition Numbers 2 / 16

Competition Graphs

Definition 1 Let D = (V, A) be a digraph. The competition graph of D is the simple graph G = (V, E) where {u, v} ∈ E if and only if N+(u) ∩ N+(v) = ∅. The competition graph of D is denoted C(D).

Jaromy Kuhl (UWF) Competition Numbers 2 / 16

Competition Graphs

Definition 1 Let D = (V, A) be a digraph. The competition graph of D is the simple graph G = (V, E) where {u, v} ∈ E if and only if N+(u) ∩ N+(v) = ∅. The competition graph of D is denoted C(D). Which graphs are competition graphs?

Jaromy Kuhl (UWF) Competition Numbers 2 / 16

Let S = {S1, . . . , Sm} be a family of cliques in a graph G.

Jaromy Kuhl (UWF) Competition Numbers 3 / 16

Let S = {S1, . . . , Sm} be a family of cliques in a graph G. S is an edge clique cover of G provided {u, v} ∈ E(G) if and only if {u, v} ⊆ Si.

Jaromy Kuhl (UWF) Competition Numbers 3 / 16

Let S = {S1, . . . , Sm} be a family of cliques in a graph G. S is an edge clique cover of G provided {u, v} ∈ E(G) if and only if {u, v} ⊆ Si. θe(G) = min{|S| : S is an edge clique cover of G}.

Jaromy Kuhl (UWF) Competition Numbers 3 / 16

Which graphs are competition graphs of acyclic digraphs?

Jaromy Kuhl (UWF) Competition Numbers 4 / 16

Which graphs are competition graphs of acyclic digraphs? For sufficiently large k, G ∪ Ik is the competition graph of an acyclic digraph.

Jaromy Kuhl (UWF) Competition Numbers 4 / 16

Which graphs are competition graphs of acyclic digraphs? For sufficiently large k, G ∪ Ik is the competition graph of an acyclic digraph. Definition 2 The competition number of G is k(G) = min{k : G ∪ Ik is the competition graph of an acyclic digraph}

Jaromy Kuhl (UWF) Competition Numbers 4 / 16

A digraph D = (V, A) is acyclic if and only if there is an ordering v1, v2, . . . , vn of the vertices in V such that if (vi, vj) ∈ A, then i < j.

Jaromy Kuhl (UWF) Competition Numbers 5 / 16

A digraph D = (V, A) is acyclic if and only if there is an ordering v1, v2, . . . , vn of the vertices in V such that if (vi, vj) ∈ A, then i < j. G is the competition graph of an acyclic digraph if and only if there is an ordering v1, . . . , vn and there is an edge clique cover {S1, . . . , Sn} such that Si ⊆ {v1, . . . , vi−1} for each i.

Jaromy Kuhl (UWF) Competition Numbers 5 / 16

Theorem 1 k(Kn,n) = n2 − 2n + 2

Jaromy Kuhl (UWF) Competition Numbers 6 / 16

Theorem 1 k(Kn,n) = n2 − 2n + 2 Theorem 2 k(Kn,n,n) = n2 − 3n + 4

Jaromy Kuhl (UWF) Competition Numbers 6 / 16

Theorem 1 k(Kn,n) = n2 − 2n + 2 Theorem 2 k(Kn,n,n) = n2 − 3n + 4 Theorem 3 For positive integers x, y and z where 2 ≤ x ≤ y ≤ z, k(Kx,y,z) =

• yz − 2y − z + 4,

if x = y yz − z − y − x + 3, if x = y

Jaromy Kuhl (UWF) Competition Numbers 6 / 16

Theorem 4 If n ≥ 5 is odd, then n2 − 4n + 7 ≤ k(K 4

n ) ≤ n2 − 4n + 8.

Theorem 5 If n is prime and m ≤ n, then k(K m

n ) ≤ n2 − 2n + 3.

Jaromy Kuhl (UWF) Competition Numbers 7 / 16

Theorem 6 k(Kn,n,n) = n2 − 3n + 4 Proof. Let L be the latin square of order n such that (a, b, c) ∈ L if and only if c ≡ a + b − 1 mod n. Consider the cliques ∆(1, 1, 1), ∆(2, n, 1), ∆(1, n, n), ∆(n, 1, n), ∆(n, 2, 1), ∆(1, 2, 2), and ∆(n − 1, 2, n), ∆(2, n − 1, n), ∆(1, n − 1, n − 1), ∆(n − 2, 2, n − 1), ∆(2, n − 2, n − 1), ∆(1, n − 2, n − 2), ...

Jaromy Kuhl (UWF) Competition Numbers 8 / 16

Consider Kx,y,z (x ≤ y ≤ z).

Jaromy Kuhl (UWF) Competition Numbers 9 / 16

Consider Kx,y,z (x ≤ y ≤ z). Definition 3 An r-multi latin square of order n is an n × n array of nr symbols such that each symbol appears once in each row and column and each cell contains r symbols.

Jaromy Kuhl (UWF) Competition Numbers 9 / 16

K2,4,6

Jaromy Kuhl (UWF) Competition Numbers 10 / 16

K2,4,6 1,2 4,5 3,7 6,8 5,6 7,8 1,2 3,4 7,8 2,3 4,6 1,5 3,4 1,6 5,8 2,7

Jaromy Kuhl (UWF) Competition Numbers 10 / 16

K2,4,6 1,2 4,5 3,7 6,8 5,6 7,8 1,2 3,4 7,8 2,3 4,6 1,5 3,4 1,6 5,8 2,7 1,2 4,5 3 6 3,4 1,6 5 2

Jaromy Kuhl (UWF) Competition Numbers 10 / 16

K2,4,6 1,2 4,5 3,7 6,8 5,6 7,8 1,2 3,4 7,8 2,3 4,6 1,5 3,4 1,6 5,8 2,7 1,2 4,5 3 6 3,4 1,6 5 2 F = {∆(1, 1, 1), ∆(1, 1, 2), ∆(1, 2, 4), ∆(1, 2, 5), ∆(1, 3, 3), ∆(1, 4, 6), ∆(2, 1, 3), ∆(2, 1, 4), ∆(2, 2, 1), ∆(2, 2, 6), ∆(2, 3, 5), ∆(2, 4, 2),

Jaromy Kuhl (UWF) Competition Numbers 10 / 16

K2,4,6 1,2 4,5 3,7 6,8 5,6 7,8 1,2 3,4 7,8 2,3 4,6 1,5 3,4 1,6 5,8 2,7 1,2 4,5 3 6 3,4 1,6 5 2 F = {∆(1, 1, 1), ∆(1, 1, 2), ∆(1, 2, 4), ∆(1, 2, 5), ∆(1, 3, 3), ∆(1, 4, 6), ∆(2, 1, 3), ∆(2, 1, 4), ∆(2, 2, 1), ∆(2, 2, 6), ∆(2, 3, 5), ∆(2, 4, 2), ∆(1, 5), ∆(1, 6), ∆(3, 1), ∆(3, 2), ∆(4, 3), ∆(4, 4), ∆(2, 2), ∆(2, 3), ∆(3, 4), ∆(3, 6), ∆(4, 1), ∆(4, 5)}

Jaromy Kuhl (UWF) Competition Numbers 10 / 16

Let q and r be positive integers such that z = qy + r.

Jaromy Kuhl (UWF) Competition Numbers 11 / 16

Let q and r be positive integers such that z = qy + r. Let L be a (q + 1)-multi latin square of order y.

Jaromy Kuhl (UWF) Competition Numbers 11 / 16

Let q and r be positive integers such that z = qy + r. Let L be a (q + 1)-multi latin square of order y. Let R′ = {r ′

i : 1 ≤ i ≤ x} be a set of rows and let S′ = {s′ i : 1 ≤ i ≤ z}

be a set of z symbols.

Jaromy Kuhl (UWF) Competition Numbers 11 / 16

Let q and r be positive integers such that z = qy + r. Let L be a (q + 1)-multi latin square of order y. Let R′ = {r ′

i : 1 ≤ i ≤ x} be a set of rows and let S′ = {s′ i : 1 ≤ i ≤ z}

be a set of z symbols. Set L(R′, C, S′) = {(r ′

i , cj, s′ k) : (r ′ i , cj, s′ k) ∈ L, r ′ i ∈ R′, s′ k ∈ S′}.

Jaromy Kuhl (UWF) Competition Numbers 11 / 16

Let q and r be positive integers such that z = qy + r. Let L be a (q + 1)-multi latin square of order y. Let R′ = {r ′

i : 1 ≤ i ≤ x} be a set of rows and let S′ = {s′ i : 1 ≤ i ≤ z}

be a set of z symbols. Set L(R′, C, S′) = {(r ′

i , cj, s′ k) : (r ′ i , cj, s′ k) ∈ L, r ′ i ∈ R′, s′ k ∈ S′}.

Lemma 1 The family F = {∆(i, j, k) : (r ′

i , cj, s′ k) ∈ L(R′, C, S′)}∪

{∆(j, k) : (ri, cj, sk) ∈ L(R \ R′, C, S′)} is an edge clique cover of Kx,y,z. Moreover, θ(Kx,y,z) = yz.

Jaromy Kuhl (UWF) Competition Numbers 11 / 16

Theorem 7 For positive integers x, y and z where 2 ≤ x ≤ y ≤ z, k(Kx,y,z) =

• yz − 2y − z + 4,

if x = y yz − z − y − x + 3, if x = y

Jaromy Kuhl (UWF) Competition Numbers 12 / 16

Theorem 7 For positive integers x, y and z where 2 ≤ x ≤ y ≤ z, k(Kx,y,z) =

• yz − 2y − z + 4,

if x = y yz − z − y − x + 3, if x = y Let L be a (q + 1)-multi latin square of order y such that (i, j, k) ∈ L if and only if i + j − 1 ≡ k mod y.

Jaromy Kuhl (UWF) Competition Numbers 12 / 16

Theorem 7 For positive integers x, y and z where 2 ≤ x ≤ y ≤ z, k(Kx,y,z) =

• yz − 2y − z + 4,

if x = y yz − z − y − x + 3, if x = y Let L be a (q + 1)-multi latin square of order y such that (i, j, k) ∈ L if and only if i + j − 1 ≡ k mod y. Let R′ = {r1, . . . , rx−1, ry} and let S′ = {s1, . . . , sz}.

Jaromy Kuhl (UWF) Competition Numbers 12 / 16

Example: x = 3, y = 5, z = 13 1,6,11 2,7,12 3,8,13 4,9,14 5,10,15 2,7,12 3,8,13 4,9,14 5,10,15 1,6,11 3,8,13 4,9,14 5,10,15 1,6,11 2,7,12 4,9,14 5,10,15 1,6,11 2,7,12 3,8,13 5,10,15 1,6,11 2,7,12 3,8,13 4,9,14

Jaromy Kuhl (UWF) Competition Numbers 13 / 16

Example: x = 3, y = 5, z = 13 1,6,11 2,7,12 3,8,13 4,9,14 5,10,15 2,7,12 3,8,13 4,9,14 5,10,15 1,6,11 3,8,13 4,9,14 5,10,15 1,6,11 2,7,12 4,9,14 5,10,15 1,6,11 2,7,12 3,8,13 5,10,15 1,6,11 2,7,12 3,8,13 4,9,14 1,6,11 2,7,12 3,8,13 4,9 5,10 2,7,12 3,8,13 4,9 5,10 1,6,11 5,10 1,6,11 2,7,12 3,8,13 4,9

Jaromy Kuhl (UWF) Competition Numbers 13 / 16

Case 1: x = y ∆1 = {u1, v1, w1}, ∆2 = {u2, vy, w1}, ∆3 = {u1, vy, wy}, ∆4 = {uy, v1, wy}, ∆5 = {uy, v2, w1}, ∆6 = {u1, v2, w2}

Jaromy Kuhl (UWF) Competition Numbers 14 / 16

Case 1: x = y ∆1 = {u1, v1, w1}, ∆2 = {u2, vy, w1}, ∆3 = {u1, vy, wy}, ∆4 = {uy, v1, wy}, ∆5 = {uy, v2, w1}, ∆6 = {u1, v2, w2} 0 ≤ s ≤ y − 4: ∆3s+7 = {uy−s−1, v2, wy−s}, ∆3s+8 = {u2, vy−s−1, wy−s}, and ∆3s+9 = {u1, vy−s−1, wy−s−1}

Jaromy Kuhl (UWF) Competition Numbers 14 / 16

Case 1: x = y ∆1 = {u1, v1, w1}, ∆2 = {u2, vy, w1}, ∆3 = {u1, vy, wy}, ∆4 = {uy, v1, wy}, ∆5 = {uy, v2, w1}, ∆6 = {u1, v2, w2} 0 ≤ s ≤ y − 4: ∆3s+7 = {uy−s−1, v2, wy−s}, ∆3s+8 = {u2, vy−s−1, wy−s}, and ∆3s+9 = {u1, vy−s−1, wy−s−1} 0 ≤ s ≤ z − y − 1: ∆3y−2+s = {u, v, wy+s+1}

Jaromy Kuhl (UWF) Competition Numbers 14 / 16

Case 2: x < y ∆1 = {ux, v1, wy}, ∆2 = {v2, wy}, ∆3 = {ux, v2, w1}, ∆4 = {u1, v1, w1}, ∆5 = {u1, v2, w2}, ∆6 = {u2, v1, w2}, ∆7 = {u2, vy, w1}

Jaromy Kuhl (UWF) Competition Numbers 15 / 16

Case 2: x < y ∆1 = {ux, v1, wy}, ∆2 = {v2, wy}, ∆3 = {ux, v2, w1}, ∆4 = {u1, v1, w1}, ∆5 = {u1, v2, w2}, ∆6 = {u2, v1, w2}, ∆7 = {u2, vy, w1} 0 ≤ s ≤ y − x − 1: ∆2s+8 = {ux, vy−s, wy−s−1}, and ∆2s+9 = {u1, vy−s−1, wy−s−1}

Jaromy Kuhl (UWF) Competition Numbers 15 / 16

0 ≤ s ≤ x − 4: ∆3s+2(y−x)+8 = {ux−s−1, v2, wx−s}, ∆3s+2(y−x)+9 = {u2, vx−s−1, wx−s}, and ∆3s+2(y−x)+10 = {u1, vx−s−1, wx−s−1} 0 ≤ s ≤ z − y − 1: ∆2y+x−1+s = {u, v, wy+s+1}

Jaromy Kuhl (UWF) Competition Numbers 16 / 16